Solve your IGNOU Doubts

Question:

Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

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Question:

Find the least value of $a^2 \sec^2 x + b^2 \csc^2 x$ , where a > 0, b > 0.

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Question:

) Express $\frac{x - 1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

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Question:

If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.

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Question:

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication $(\cdot)$ is given. Define (*) on $\mathbf{R}$ as:

$$a <em> b = \frac{a + b}{2}, \forall a, b \in \mathbf{R}.$
Is () associative in $\mathbf{R}$ ? Is $(\cdot)$ distributive over (*) in $\mathbf{R}$ ? Check.

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Question:

Find the domain of the function f given by $f(x) = \sqrt{\frac{2 - x}{x^2 + 1}}$ .

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Question:

$a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

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Question:

Every integrable function is monotonic.

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Question:

) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .

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Question:

The greatest interger function is continuous on $\mathbf{R}$ .

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Question:

The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.

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Question:

. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate. $\quad (10)$